1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. 285

4.3. Fary functors

Up until now, we have been developing a kind of a calculus of Euler characteristics.

In particular, for a constructible function f : X ----> Z, and a sufficiently nice pair of

compact subsets Y :J Z of X, we have the Euler characteristic x(Y, Z; f). In this
section, we introduce the notion of a Fary functor. This is an object that assigns to
the pair (Y, Z) a collection of homology groups Hi(Y, Z; F). These groups satisfy
a set of properties similar to the Eilenberg-Steenrod axioms. So you may think of
a Fary functor as a "homology theory" on the space X. Even though we will not
have time for many examples, Fary functors arise quite often in geometry. (For the
reader familiar with the theory of constructible sheaves: a Fary functor is a model
of a constructible complex of sheaves. More precisely, any constructible complex
gives rise to a Fary functor.) As we plunge into the definition, the example to keep
in mind is the ordinary homology Fary functor Z, which assigns to the pair (Y, Z)
the ordinary homology of Y relative to Z.
Let X be a real algebraic manifold, with a fixed semi-algebraic stratification S.
A Fary functor F on X does the following three things.

(1) It assigns to each standard pair (Y, Z) in X, and each i E Z, a finitely
generated abelian group Hi(Y, Z; F), which is non-zero for finitely many i.

(2) It gives a push-forward map Hi(Y, Z; F)----> Hi(Y', Z'; F), whenever R(Y')n

Y c R(Y) and G(Y) n Y' c G(Y'). Here we let G(Y) = Z, R(Y) = 8Y \ Z, and

similarly for Y'.

(3) It gives a boundary map 8 : Hi(Y, Z; F) ----> Hi-i(Y', Z'; F), whenever

G(Y) c R(Y').

The structures (1)-(3) are subject to the following four axioms.

(Ax 1) Functoriality. The map H*(Y, Z; F) ----> H *(Y, Z; F) is the identity.

Given any three standard pairs {(Yk, Zk)H=I such that the push-forward maps

fk , 1 : H*(Yki Zk; F)----> H.(Yi, Z1; F) are defined fork< l, we have h,3 o !i,2 = fi,3.

(Ax 2) Naturality of 8. The boundary maps commute with the push-forward
maps whenever all are defined.

(Ax 3) The long exact sequence. Assume we have three standard pairs (Y, Z),

(Y', Z'), (Y", Z"), such that:

G(Y) = G(Y'), R(Y') = R(Y"), and G(Y") = R(Y).

Then the sequence:

is exact.

· · · ____, Hi+1 (Y", Z"; F) ____,

Hi (Y, Z; F) ____, Hi (Y', Z'; F) ____, Hi (Y", Z"; F) ____,

Hi-I (Y, Z; F) ____, ...

(Ax 4) Homotopy. This consists of two parts:

(a) Suppose (yt, Zt), t E [O, 1], is a smoothly varying family of standard pairs,
such that the push-forward H (Yi, Zt; F) ____, H (yt, , Zt'; F) is defined for all t < t'.